Group #20299412 — subgroups

Subgroups — #20299412

Order 7232.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813669	C₂	1	2	yes	maximal, complemented
2	#16813652	C₂	1	2	yes	maximal, complemented
3	#16813651	C₂	1	4	yes	maximal, complemented
4	#16813641	C₂	1	2	yes	maximal, complemented
5	C₂×Dic₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×D₁₁₃	C₂	1	2	yes	maximal, complemented
7	#16813628	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#14268647	—	2	2	no	—
10	#14268647	C₂²	1	1	yes	characteristic
11	#14268646	—	2	4	no	—
12	#14268646	C₂²	1	4	yes	—
13	C₂×Dic₂₂₆	—	2	2	no	—
14	C₂×Dic₂₂₆	C₂²	1	2	yes	—
15	Q₈×D₁₁₃	C₂²	1	4	yes	—
16	Dic₄₅₂	C₂²	1	2	yes	—
17	Dic₄₅₂	C₂²	1	2	yes	—
18	#14268637	C₂²	1	4	yes	—
19	#14268635	C₂²	1	4	yes	—
20	#14268631	C₂²	1	1	yes	characteristic
21	#14268630	C₂²	1	2	yes	—
22	C₈×D₁₁₃	C₂²	1	2	yes	—
23	#41×C₁₁₃	C₂²	1	2	yes	—
24	Q₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
25	SD₁₆×C₁₁₃	C₂²	1	2	yes	—
26	D₈×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
28	Q₈×C₁₁₃	C₂³	1	2	yes	—
29	D₄×C₁₁₃	—	2	2	no	—
30	D₄×C₁₁₃	C₂³	1	2	yes	—
31	C₂×C₄×C₁₁₃	—	2	2	no	—
32	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₁₃	—	2	4	no	—
34	C₂²⋊D₁₁₃	D₄	1	2	yes	—
35	C₂×Dic₁₁₃	—	2	2	no	—
36	C₂×Dic₁₁₃	—	2	4	no	—
37	D₄₅₂	D₄	1	1	yes	characteristic
38	C₄×D₁₁₃	—	2	4	no	—
39	C₄×D₁₁₃	C₂³	1	2	yes	—
40	Dic₂₂₆	—	2	2	no	—
41	Dic₂₂₆	—	2	4	no	—
42	Dic₂₂₆	D₄	1	1	yes	characteristic
43	Dic₂₂₆	C₂³	1	4	yes	—
44	C₈×C₁₁₃	C₂³	1	1	yes	characteristic
45	C₈×C₁₁₃	C₂³	1	1	yes	characteristic
46	#11744751	C₂³	1	2	yes	—
47	C₂²×C₁₁₃	—	2	2	no	—
48	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₁₃	C₂×D₄	1	2	yes	—
50	C₄×C₁₁₃	—	2	2	no	—
51	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
52	C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
53	Dic₁₁₃	—	2	4	no	—
54	Dic₁₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₁₁₃	—	2	1	no	—
56	C₂×C₁₁₃	—	4	2	no	—
57	C₂×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
58	D₁₁₃	—	2	2	no	—
59	C₁₁₃	#578	1	1	yes	characteristic, complemented, Sylow
60	#578	—	113	1	no	maximal, Sylow
61	2₋¹⁺⁴	—	113	2	no	—
62	#137	—	113	2	no	—
63	#137	—	113	4	no	—
64	#135	—	113	2	no	—
65	#135	D₁₁₃	1	1	yes	characteristic, complemented
66	C₂×Q₁₆	—	113	2	no	—
67	C₂×Q₁₆	—	113	1	no	—
68	#131	—	113	1	no	—
69	#41	—	113	1	no	—
70	#41	—	113	4	no	—
71	#41	—	226	2	no	—
72	#41	—	226	4	no	—
73	#41	C₂×D₁₁₃	1	2	yes	—
74	C₂×Q₈	—	113	4	no	—
75	C₂×Q₈	—	113	2	no	—
76	C₂×Q₈	—	226	2	no	—
77	Q₁₆	—	113	4	no	—
78	Q₁₆	—	113	2	no	—
79	Q₁₆	—	113	2	no	—
80	Q₁₆	C₂×D₁₁₃	1	1	yes	characteristic
81	SD₁₆	—	113	4	no	—
82	SD₁₆	C₂×D₁₁₃	1	2	yes	—
83	D₈	C₂×D₁₁₃	1	1	yes	characteristic
84	M₁₆	—	113	1	no	—
85	M₁₆	—	113	2	no	—
86	C₂×C₈	—	113	2	no	—
87	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
88	Q₈	—	113	1	no	—
89	Q₈	—	113	4	no	—
90	Q₈	—	226	4	no	—
91	Q₈	—	226	2	no	—
92	Q₈	C₂²×D₁₁₃	1	2	yes	—
93	D₄	—	2	2	no	—
94	D₄	—	113	2	no	—
95	D₄	—	113	1	no	—
96	D₄	—	226	4	no	—
97	D₄	C₂²×D₁₁₃	1	2	yes	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	113	2	no	—
100	C₂×C₄	—	226	4	no	—
101	C₂×C₄	—	226	4	no	—
102	C₂×C₄	—	226	2	no	—
103	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
104	C₈	—	113	2	no	—
105	C₈	C₂²×D₁₁₃	1	1	yes	characteristic
106	C₈	C₂²×D₁₁₃	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	113	2	no	—
109	C₂²	D₄×D₁₁₃	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	113	2	no	—
112	C₄	—	226	4	no	—
113	C₄	D₄×D₁₁₃	1	1	yes	characteristic
114	C₄	C₂³×D₁₁₃	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	4	2	no	—
117	C₂	—	226	2	no	—
118	C₂	C₂×D₄×D₁₁₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813669

C₂

yes

maximal, complemented

#16813652

C₂

yes

maximal, complemented

#16813651

C₂

yes

maximal, complemented

#16813641

C₂

yes

maximal, complemented

C₂×Dic₄₅₂

C₂

yes

maximal, characteristic, complemented

Q₁₆×D₁₁₃

C₂

yes

maximal, complemented

#16813628

C₂

yes

maximal, characteristic, complemented

#135×C₁₁₃

C₂

yes

maximal, characteristic, complemented

#14268647

—

#14268647

C₂²

yes

characteristic

#14268646

—

#14268646

C₂²

yes

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

C₂²

yes

—

Q₈×D₁₁₃

C₂²

yes

—

Dic₄₅₂

C₂²

yes

—

Dic₄₅₂

C₂²

yes

—

#14268637

C₂²

yes

—

#14268635

C₂²

yes

—

#14268631

C₂²

yes

characteristic

#14268630

C₂²

yes

—

C₈×D₁₁₃

C₂²

yes

—

#41×C₁₁₃

C₂²

yes

—

Q₁₆×C₁₁₃

C₂²

yes

characteristic

SD₁₆×C₁₁₃

C₂²

yes

—

D₈×C₁₁₃

C₂²

yes

characteristic

C₂×C₈×C₁₁₃

C₂²

yes

characteristic

Q₈×C₁₁₃

C₂³

yes

—

D₄×C₁₁₃

—

D₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂²⋊D₁₁₃

—

C₂²⋊D₁₁₃

D₄

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

D₄₅₂

D₄

yes

characteristic

C₄×D₁₁₃

—

C₄×D₁₁₃

C₂³

yes

—

Dic₂₂₆

—

Dic₂₂₆

—

Dic₂₂₆

D₄

yes

characteristic

Dic₂₂₆

C₂³

yes

—

C₈×C₁₁₃

C₂³

yes

characteristic

C₈×C₁₁₃

C₂³

yes

characteristic

#11744751

C₂³

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

characteristic

C₂×D₁₁₃

C₂×D₄

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₄×C₁₁₃

C₂⁴

yes

characteristic

Dic₁₁₃

—

Dic₁₁₃

C₂×D₄

yes

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×D₄

yes

characteristic

D₁₁₃

—

C₁₁₃

#578

yes

characteristic, complemented, Sylow

#578

—

113

maximal, Sylow

2₋¹⁺⁴

—

113

—

#137

—

113

—

#137

—

113

—

#135

—

113

—

#135

D₁₁₃

yes

characteristic, complemented

C₂×Q₁₆

—

113

—

C₂×Q₁₆

—

113

—

#131

—

113

—

#41

—

113

—

#41

—

113

—

#41

—

226

—

#41

—

226

—

#41

C₂×D₁₁₃

yes

—

C₂×Q₈

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

Q₁₆

—

113

—

Q₁₆

—

113

—

Q₁₆

—

113

—

Q₁₆

C₂×D₁₁₃

yes

characteristic

SD₁₆

—

113

—

SD₁₆

C₂×D₁₁₃

yes

—

D₈

C₂×D₁₁₃

yes

characteristic

M₁₆

—

113

—

M₁₆

—

113

—

C₂×C₈

—

113

—

C₂×C₈

C₂×D₁₁₃

yes

characteristic

Q₈

—

113

—

Q₈

—

113

—

Q₈

—

226

—

Q₈

—

226

—

Q₈

C₂²×D₁₁₃

yes

—

D₄

—

D₄

—

113

—

D₄

—

113

—

D₄

—

226

—

D₄

C₂²×D₁₁₃

yes

—

C₂×C₄

—

C₂×C₄

—

113

—

100

C₂×C₄

—

226

—

101

C₂×C₄

—

226

—

102

C₂×C₄

—

226

—

103

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

104

C₈

—

113

—

105

C₈

C₂²×D₁₁₃

yes

characteristic

106

C₈

C₂²×D₁₁₃

yes

characteristic

107

C₂²

—

108

C₂²

—

113

—

109

C₂²

D₄×D₁₁₃

yes

characteristic

110

C₄

—

111

C₄

—

113

—

112

C₄

—

226

—

113

C₄

D₄×D₁₁₃

yes

characteristic

114

C₄

C₂³×D₁₁₃

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

226

—

118

C₂

C₂×D₄×D₁₁₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂⁴
Frattini	C₄	C₂³×D₁₁₃
Fitting	#135×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₂²×D₄
hypercenter	#135	D₁₁₃
hypocenter	C₁₁₃	#578
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₂

C₂×D₄×D₁₁₃

commutator

C₄×C₁₁₃

C₂⁴

Frattini

C₄

C₂³×D₁₁₃

Fitting

#135×C₁₁₃

C₂

socle

C₂×C₁₁₃

C₂²×D₄

D₁₁₃

C₁₁₃

G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #16813669	C₂	semidirect
2. #16813652	C₂	semidirect
3. #16813651	C₂	semidirect
4. #16813641	C₂	semidirect
5. C₂×Dic₄₅₂	C₂	semidirect
6. Q₁₆×D₁₁₃	C₂	semidirect
7. #16813628	C₂	semidirect
8. #135×C₁₁₃	C₂	semidirect
59. C₁₁₃	#578	semidirect
65. #135	D₁₁₃	semidirect

H ≅ G/N

type

1. #16813669

C₂

semidirect

2. #16813652

C₂

semidirect

3. #16813651

C₂

semidirect

4. #16813641

C₂

semidirect

5. C₂×Dic₄₅₂

C₂

semidirect

6. Q₁₆×D₁₁₃

C₂

semidirect

7. #16813628

C₂

semidirect

8. #135×C₁₁₃

C₂

semidirect

59. C₁₁₃

#578

semidirect

65. #135

D₁₁₃

semidirect

Subgroups — #20299412

Conjugacy classes of subgroups

Complements